History
A 1956 Gazette article with zero citations anticipated spacetime crystallography by six decades.
Fletcher, 1956
Trevor J. Fletcher — then a lecturer at Sir John Cass College, London, later one of Britain's best-known mathematics educators — made 16 mm mathematical animation films by hand: Plucked Strings (1954), The Cardioid, Four Line Conics (produced at the National Film Board of Canada, six festival awards). Drawing frames is expensive, and Fletcher noticed that the symmetries of a cyclically repeating diagram let one drawing serve several moments of the film: "the symmetries present in a diagram can often be used to reduce the labour of drawing." His five-page article Film Groups (Math. Gazette 40, 15–19, Feb. 1956) then does something remarkably modern: it treats a looping film as a static three-dimensional pattern in coordinates (t, y, z), applies the classification of space groups, and strikes out every group that "would interchange time and space axes". For animations both looping and doubly periodic in space he counts, from the 230 space groups minus the 36 cubic ones, 194 film groups, his name for them. He knew exactly what a time glide was — "reflection together with a shift in time… then the same drawing can be used again (turned if necessary) at different stages in the cycle" — and noted that the diagrams of Plucked Strings realise the group numbered 26 in the strip-group tables of Bhagavantam–Venkatarayudu, his reference [1]. He even analysed strobing: a rigid drawing of cos(ny − mx) pulled past a shutter produces apparent transverse motion, his diagnosis of the Dance–Kaufmann wave films, complete with a Zoetrope recipe.
Semantic Scholar lists zero citations for the article, and OpenCitations returns none. It was, as far as we can tell, never followed up.
What Fletcher got wrong (instructively)
Two conventions separate his counts from the modern ones.
7 vs 13 (strips). For patterns with one space and one time dimension Fletcher took the seven primitive rectangular-and-oblique wallpaper groups (p1, p2, pm, pg, pmm, pmg, pgg) of the (x,t)-plane. Two conventions cost him six groups. First, wallpaper equivalence does not care which axis is which, but spacetime does: reflection across the strip (mx) and reflection of time (mt) are different operations, so pm, pg and pmg each split in two. Second, he excluded the centred groups as "based on rhombic lattices" — but a centred spacetime lattice is a perfectly realisable animation in which neighbours run half a period out of phase, and cm itself splits (Cmx, Cmt) while cmm gives one more: 7 + 3 + 3 = 13, first counted by Xu–Wu (2018).
194 vs 275 (the plane case). In the static (t,y,z) reading every spacetime-group operation — including time reversal, which becomes the mirror perpendicular to the t-axis, and glide time-reversal, an ordinary glide plane — is a 3D space-group operation, so each spacetime group is one of the 230 (never cubic: we verified all 275 against standard space-group tables). Fletcher's subtraction fails in the other direction: the map is many-to-one. A single 3D space-group type can sit in spacetime in several inequivalent ways, because the time axis must be chosen and different invariant directions are genuinely different animations — the monoclinic system splits into T- and R-monoclinic, the base-centred orthorhombic lattice into two types, and mirror-versus-time-reversal readings of one 3D reflection give distinct groups. Correctly counted, via the group-cohomology classification of Xu–Wu (2018, appendix C), the total is 275, verified independently by the computation in this repository.
The other ancestors
Shubnikov's antisymmetry (1945–). The "prime" operation of black-white symmetry was originally interpreted as time reversal; Zamorzaev's 1953 thesis derived the 1651 antisymmetry space groups, and his Chișinău school's P-symmetry generalised colour to arbitrary quality groups. A looping N-frame animation is literally a ZN colour symmetry — "time as colour" is the discrete shadow of the spacetime group's phase circle, and our tilde-marked mirrors obey the same parity rules as the 46 two-colour wallpaper groups.
Janssen–Janner–Ascher (1969). "Crystallographic groups in space and time" set up the general theory (including the central-extension machinery) that the modern physics literature builds on; the 2+1D enumeration seems not to have been carried out then.
Choreographic crystals and the H/K theorem. For finite configurations, Boyle–Khoo–Smith (2016) classified "choreographic order" (satellite swarms whose symmetries are space ops with time offsets); in equivariant dynamics the Golubitsky–Stewart H/K theorem describes exactly which pairs (setwise, pointwise) symmetry groups a time-periodic state can have — the point-group shadow of a spacetime group's clock character.
Conway's program. Orbifold notation for wallpaper groups (with Thurston, 1980s; Conway–Huson 2002), colour decorations in The Symmetries of Things (2008), fibrifold names for the space groups (Conway–Delgado‑Friedrichs–Huson–Thurston 2001), and — months before this site — Conway-type orbifold symbols for the 80 layer groups (Mahmoudi et al., Dec. 2025). Our clockwork decorations extend the same program along the time direction; to our knowledge no time-decorated orbifold notation existed before.
Fletcher's closing line
He ends with Klein: geometry is the study of invariants of a group, and "mathematical film making is both a science and an art. The symmetry groups are the key to both the structure of the science and the aesthetics of the art." This site is an attempt to build the catalog he was one abstraction away from writing.
Sources
Fletcher, Film Groups, Math. Gazette 40 (1956) 15–19, doi:10.2307/3610262 · ATM's digitised Fletcher films · Xu–Wu (the 275), arXiv:1703.03388 · Ke–Wu (later re-derivation), arXiv:2604.05619 · Janssen–Janner–Ascher, Physica 41 (1969) 541; 42 (1969) 41, 71 · Boyle–Khoo–Smith, arXiv:1407.5876 · Golubitsky–Stewart, The Symmetry Perspective (2002) · CDHT, arXiv:math/9911185 · Mahmoudi–Dresselhaus–Dimitriyev, arXiv:2512.05149 · Litvin, Magnetic Group Tables.